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1. (1) Natural Numbers: All the counting numbers are called natural number.
Example: 1, 2, 3, 4, 5, ...
(a) Even Numbers: The numbers which are exactly divisible by 2 are called even
numbers. Example: 2, 4, 6, 8, ...
(b) Odd Numbers: The numbers which leave a remainder 1 when divided by 2 are called
odd numbers. Example: 1, 3, 5, 7, ....
(c) Prime Numbers: If a number is not divisible by any other number except 1 and itself,
it is called a prime number. Example: 2, 3, 5, 7, 11, ....
Co-primes →Two numbers which have no common factor between them except 1
are said to be co-prime to each other. The two numbers individually may be
prime or composite. Example: 13 and 29 are co-primes.
(d) Composite Numbers: Numbers which are divisible by other numbers along with 1
and itself are called composite numbers. Example: 4, 6, 8, 9, 10, ..... The number 1
is neither prime nor composite.

2. (2) Whole Numbers: Natural numbers along with '0' form the
set of whole numbers. Example: 0, 1, 2, 3, .....
(3) Integers: All counting numbers and their negatives along with
zero are called Integers. Example: .. .-4, -3, -2, -1, 0, 1, 2, 3, 4, ..
(4) Rational and Irrational Numbers: Any number which can be
expressed in the form of p/q, where p and q are integers and q is not
equal to 0 is a rational number. Example: 3/5, 4, -6, etc.
Numbers which are represented by non-terminating and non-recurring
decimals are called irrational numbers. Example: √√2=1.414.....
√√3=1.732.....

3. (5) Real Numbers: Rational and irrational number taken together are called real
numbers.

4. 1. a²-b² = (a + b) (a - b)
2. (a + b)² = a² + b² + 2ab
3. (a - b)²=a² + b² - 2ab
4. (a+b+c)² = a² + b² + c² + 2ab + 2bc + 2ac
5. (a+b)³ = a³ + b³ + 3ab (a + b)
6. (a -b)³=a³-b³- 3ab (a - b)
7. a³ + b³ = (a + b) (a² + b²-ab)
8. a3 -b³= (a - b) (a² + b² + ab)
9. a³ + b³ + c³-3abc = (a + b + c) (a² + b² + c²- ab- bc- ac)

5. Divisibility by 2: A number is divisible by 2 if its unit digit is zero or an even number.
Example: 248, 130.
Divisibility by 3: A number is divisible by 3 if the sum of its digit is divisible by 3.
Example: 279 -- 2+7+9 = 18. 18 is divisible by 3, hence 279 is divisible by 3.
Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4. Example: 236784,
here, 84 is divisible by 4, hence 236784 is divisible by 4.
Divisibility by 5: A number is divisible by 5 if the number or its unit digit is either 5 or 0.
Example: 115, 240 etc.
Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3. Example: 318, 396, etc.
Divisibility by 8: A number is divisible by 8 if the number formed by its last 3 digit is divisible by 8.
Example: 23816. Here, 816 is divisible by 8, hence 23816 is divisible by 8.
Divisibility by 9: A number is divisible by 9 if the sum of all its digits is divisible by 9. Example: 72936 --
>7+2+9+3+6=27